Production Costing (Chapter 8 of W&W)


 Emil Reynolds
 3 years ago
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1 Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary to run th thrmal plants. A production cost program, also rfrrd to as a production cost modl, is widly usd throughout th lctric powr industry for many purposs: Longrang systm planning: Hr, it is usd to simulat a singl futur yar following th plannd xpansion. For xampl, th Midwst ISO usd a production cost program to undrstand th ffct on nrgy prics of building HVC from th Midwst US to th East coast. Ful budgting: Many companis run production cost programs to dtrmin th amount of natural gas and coal thy will nd to purchas in th coming wks or months. Maintnanc: Production cost programs ar run to dtrmin maintnanc schduls for gnration. Enrgy intrchang: Production cost programs ar run to facilitat ngotiations for nrgy intrchang btwn companis.
2 Thr ar two ssntial inputs for any production cost program:. ata charactrizing futur load 2. ata charactrizing gnration costs, in trms of: a. Hat rat curvs and b. Ful costs All production cost programs rquir at last th abov data. Spcific programs will rquir additional data dpnding on thir particular dsign. Th information providd by production costing includs th annual costs of oprating th gnration facilitis, a cost that is dominatd by th ful costs but also affctd by th maintnanc costs. Production costing may also provid mor timgranular stimats of ful and maintnanc costs, such as monthly, wkly, or hourly, from which it is thn possibl to obtain annual production costs. A simplifid way to considr a production cost program is as an hourbyhour simulation of th powr systm ovr a duration of T hours, whr at ach hour, Th load is spcifid; A unit commitmnt dcision is mad; A dispatch dcision is mad to obtain th production costs for that hour 2
3 Th total production costs is thn th sum of hourly production costs ovr all hours,,t. Som production programs do in fact simulat hourbyhour opration in this mannr. An important charactrizing fatur is how th program maks th unit commitmnt (UC) and dispatch dcisions. Th simplst approach maks th UC dcision basd on priority ordring such that units with lowst avrag cost ar committd first. Startup costs ar addd whn a unit is startd, but thos costs do not figur into th optimization. Th simplst approach for making th dispatch dcision is rfrrd to as th block loading principl, whr ach unit committd is fully loadd bfor th nxt unit is committd. Th last unit is dispatchd at that lvl ncssary to satisfy th load. Gratr lvls of sophistication may b mbddd in production cost programs, as dscribd blow: Unit commitmnt and dispatch: A full unit commitmnt program may b run for crtain blocks of intrvals at a tim,.g., a wk. Hydro: Hydrothrmal coordination may b implmntd. 3
4 Ntwork rprsntation: Th ntwork may b rprsntd using C flow and branch limits. Locational marginal prics: LMPs may b computd. Maintnanc schduls: Maintnanc schduls may b takn into account. Uncrtainty: Load uncrtainty and gnrator unavailability may b rprsntd using probabilistic mthods. This allows for computation of rliability indics such as loss of load probability (LOLP) and xpctd unsrvd nrgy (EUE). Scurity constraints may b imposd using LOFs. Blow ar som slids that Midwst ISO uss to introduc production cost modls. What is a Production Cost Modl? Capturs all th costs of oprating a flt of gnrators Originally dvlopd to manag ful invntoris and budgt in th mid 970 s vlopd into an hourly chronological scurity constraind unit commitmnt and conomic dispatch simulation Minimiz costs whil simultanously adhring to a wid varity of oprating constraints. Calculat hourly production costs and locationspcific markt claring prics. 4 4
5 What Ar th Advantags of Production Cost Modls? Allows simulation of all th hours in a yar, not ust pak hour as in powr flow modls. Allows us to look at th nt nrgy pric ffcts through LMP s and its componnts. Production cost. Enabls th simulation of th markt on a forcast basis Allows us to look at all control aras simultanously and valuat th conomic impacts of dcisions. 5 isadvantags of Production Cost Modls Rquir significant amounts of data Long procssing tims Nw concpt for many Stakholdrs Rquir significant bnchmarking Tim consuming modl building procss Linkd to powr flow modls o not modl rliability to th sam xtnt as powr flow 6 5
6 Production Cost Modl vs. Powr Flow Production Cost Modl Powr Flow SCUC&E: vry dtaild Hand dispatch (mrit Ordr) All hours C Transmission Slctd scurity constraints Markt analysis/ Transmission analysis/planning On hour at a tim AC and C Larg numbrs of scurity constraints Basis for transmission rliability & oprational planning Commrcial grad production costing tools W will dscrib in mor dtail th construction of production costing programs latr. Hr w simply mntion som of th commrcially availabl production costing tools. Th Vntyx product Promod incorporats dtails in gnrating unit oprating charactristics, transmission grid topology and constraints, unit commitmnt/ oprating conditions, and markt systm oprations. Promod can oprat on nodal or zonal mods dpnding on th scop, timfram, and simulation rsolution of th problm. Promod is not a forcasting modl and dos not considr th pric and availability of othr fuls. 6
7 Th AICA product GTMAX, dvlopd by Argonn National Labs, can b mployd to prform rgional powr grad or national powr dvlopmnt analysis. GTMax will valuat systm opration, dtrmin optimal location of powr sourcs, and assss th bnfits of nw transmission lins. GTMax can simulat complx lctric markt and oprating issus, for both rgulatd and drgulatd markt. Th PowrCost, Inc. product GnTradr mploys conomic unit dispatch logic to analyz conomics, uncrtainty, and risk associatd with individual gnration rsourcs and portfolios. GnTradr dos not rprsnt th ntwork. PROSYM is a multiara lctric nrgy production simulation modl dvlopd by Hnwood nrgy Inc. It is an hourly simulation ngin for lastcost optimal production dispatch basd on th rsourcs marginal costs, with full rprsntation of gnrating unit charactristics, ntwork ara topology and lctrical loads. PROSYM also considrs and rspcts oprational and chronological constraints; such as minimum up and down tims, random forcd outags and transmission capacity. It is dsignd to dtrmin th station gnration, missions and 7
8 conomic transactions btwn intrconnctd aras for ach hour in th simulation priod. ABB producd th softwar calld GridViw, illustratd blow []. PLEXOS, from Plxos Solutions, is a vrsatil softwar systm that prforms production cost simulation and othr functions. It is intrsting to not that Global Enrgy Solutions (GES) in 2002 purchasd Hnwood Associats (ownr of Prosym), thn Vntyx (ownrs of Promod) purchasd GES in 2008, thn ABB (ownrs of Gridviw) purchasd Vntyx. At som point, Mark Hnwood wnt to work for Plxos Solutions (s [2]). Enrgy Exmplar now owns Plxos. 8
9 Load (MW) 3.0 Probability modls Ky to us of production cost modls is th ability to rprsnt uncrtainty in load and in gnration availability. 3. Load duration curvs A critical issu for planning is to idntify th total load lvl for which to plan. On xtrmly usful tool for doing this is th socalld load duration curv, which is formd as follows. Considr that w hav obtaind, ithr through historical data or through forcasting, a plot of th load vs. tim for a priod T, as shown in Fig. 3 blow. Tim Fig. 3: Load curv (load vs. tim) Of cours, th data charactrizing Fig. 3 will b discrt, as illustratd in Fig. 4. T 9
10 Load (MW) Load (MW) Tim Fig. 4: iscrtizd Load Curv W now divid th load rang into intrvals, as shown in Fig. 5. T Tim Fig. 5: Load rang dividd into intrvals This provids th ability to form a histogram by counting th numbr of tim intrvals containd in ach load rang. In this xampl, w assum that loads in Fig. 5 at th lowr nd of th rang ar in th rang. Th histogram for Fig. 5 is shown in Fig. 6. T 0
11 Probability Count Load (MW) Fig. 6: Histogram Figur 6 may b convrtd to a probability mass function, pmf, (which is th discrt vrsion of th probability dnsity function, pdf) by dividing ach count by th total numbr of tim intrvals, which is 23. Th rsulting plot is shown in Fig Load (MW) Fig. 7: Probability mass function Lik any pmf, th summation of all probability valus should b, which w s by th following sum: =0.999 (It is not xactly.0 bcaus thr is som rounding rror). Th probability mass function provids us with th ability to comput th probability of th load bing within a rang according to:
12 Pr (Load within Rang) Pr( Load L) (2) L in Rang W may us th probability mass function to obtain th cumulativ distribution function (CF) as: Pr (Load Valu) Pr( Load L) (3) From Fig. 7, w obtain: Pr(Load ) Pr(Load 2) Pr(Load 3) Pr(Load 4) Pr(Load 5) L 5 L Valu L L 2 L 3 L 4 Pr( Load Pr( Load Pr( Load Pr( Load Pr( Load L) L).0 L).0 L).0 L) Pr(Load 7) Pr(Load 6) L 6 Pr( Load L) L 6 Pr (Load 8) Pr( Load L) L 8 Pr (Load 9) Pr( Load L) L 9 Pr (Load 0) Pr( Load L) L 0 Pr( Load L) 0 2
13 L (MW) Probability(Load > L) Plotting ths valus vs. th load rsults in th CF of Fig L (MW) Fig. 8: Cumulativ distribution function Th plot of Fig. 8 is oftn shown with th load on th vrtical axis, as givn in Fig Probability(Load > L) Fig. 9: CF with axs switchd If th horizontal axis of Fig. 9 is scald by th tim duration of th intrval ovr which th original load 3
14 L (MW) data was takn, T, w obtain th load duration curv. This curv provids th numbr of tim intrvals that th load quals, or xcds, a givn load lvl. For xampl, if th original load data had bn takn ovr a yar, thn th load duration curv would show th numbr of hours out of that yar for which th load could b xpctd to qual or xcd a givn load lvl, as shown in Fig. 0a Numbr of hours that Load > L Fig. 0a: Load duration curv Load duration curvs ar usful in a numbr of ways. Thy provid guidanc for udging diffrnt altrnativ plans. On plan may b satisfactory for loading lvls of 90% of pak and lss. On ss from Fig. 0a that such a plan would b unsatisfactory for 438 hours pr yar (5% of th tim). Thy idntify th bas load. This is th valu that th load always xcds. In Fig. 0a, this valu is 5 MW. In Fig. 0b, which shows th LC for th 2003 MISO rgion, th valu is 40GW. 4
15 Load(GW) Pak load 25% highr than 95% load lvl % Load Lvl Numbr of Hours Fig. 0b: MISO LC for 2003 Thy provid convnint calculation of nrgy, sinc nrgy is ust th ara undr th load duration curv. For xampl, Fig. shows th ara corrsponding to th bas load nrgy consumption, which is 5MWx8760hr=43800 MWhrs. 5
16 L (MW) L (MW) Numbr of hours that Load > L Fig. : Ara corrsponding to bas load nrgy consumption Thy allow illustration of gnration commitmnt policis and corrsponding yarly unit nrgy production, as shown in Fig 2, whr w s that th nuclar plant and coal plant # ar bas loadd plants, supplying MWhrs and 7520 MWhrs, rspctivly. Coal plant #2 and NGCC plant # ar th midrang plants, and CT # is a pakr. CT # NGCC # Coal plant #2 Coal plant # Nuclar plant Numbr of hours that Load > L Fig. 2: Illustration of Unit commitmnt policy 6
17 Load duration curvs ar also usd in rliability and production costing programs in computing diffrnt rliability indics, as w will s in Sctions 4 and Gnration probability modls W considr that gnrators oby a twostat modl, i.., thy ar ithr up or down, and w assum that th procss by which ach gnrator movs btwn stats is Markov, i.., th probability distribution of futur stats dpnds only on th currnt stat and not on past stats, i.., th procss is mmorylss. In this cas, it is possibl to show that unavailability (or forcd outag rat, FOR) is th stadystat (or longrun) probability of a componnt not bing availabl and is givn by U q (4) and th availability is th longrun probability of a componnt bing availabl and is givn by A p (5) whr λ is th failur rat and μ is th rpair rat. S for complt drivation of ths xprssions. 7
18 Substituting λ=/mttf and μ=/mttr, whr MTTF is th man tim to failur, and MTTR is th man tim to rpair, w gt that MTTR U q MTTF MTTR (6) MTTF A p MTTF MTTR (7) Th probability mass function rprsnting th outagd capacity (8a) or availabl capacity (8b) corrsponding to unit is thn givn as f (d ), xprssd as f ( d ( d ) p ( d ) q ( d C ) (8a) ) q ( d ) p ( d C ) f (8b) and illustratd by Fig. 3 (w will us thm both). f (d ) A =p f (d ) A =p U =q U =q 0 C Outagd capacity, d 0 C Availabl capacity, d Fig. 3: Two stat gnrator outag modl Unavailability U xprsss th fraction of tim (not including maintnanc tim) th gnrator has bn forcd out of srvic. Availability A is th fraction of tim (not including maintnanc tim) th gnrator is availabl for srvic. U+A=. 8
19 4.0 Prliminary dfinitions Lt s charactriz th load shap curv with t=g(d), as illustratd in Fig. 4. It is important to not that th load shap curv charactrizs th (forcastd) futur tim priod and is thrfor a probabilistic charactrization of th dmand. t T t=g(d) d max Hr: d is th systm load Fig. 4: Load shap t=g(d) mand, d (MW) t is th numbr of tim units in th intrval T for which th load quals or xcds d and is most typically givn in hours or days t=g(d) xprsss functional dpndnc of t on d 9
20 T rprsnts, most typically, a yar but can b any intrval of tim (wk, month, sason, yars). Th cumulativ distribution function (cdf) is givn by t g( d) F ( d) P( d) T T (9) On may also comput th total nrgy E T consumd in th priod T as th ara undr th curv, i.., E T (0) Th avrag dmand in th priod T is obtaind from d d max max davg ET g( ) d F ( ) d () T T 0 0 Now assum th plannd systm gnration capacity, i., th installd capacity, is C T, and C T <d max. This is an undsirabl situation, sinc w will not b abl to srv som dmands, vn whn thr is no capacity outag! Nonthlss, it srvs wll to undrstand th rlation of th load duration curv to svral usful indics. Th situation is illustratd in Fig. 5. d max 0 g ( ) dλ 20
21 t T t=g(d) t C C T d max mand, d (MW) Fig. 5: Illustration of Unsrvd mand Thn, undr th assumption that th givn capacity C T is prfctly rliabl, w may xprss thr usful rliability indics: Loss of load xpctation, LOLE: th xpctd numbr of tim units that load will xcd capacity LOLE t g C ) (2) T C ( T Loss of load probability, LOLP: th probability that th dmand will qual or xcd capacity during T: LOLP P C ) F ( C ) (3) ( T T W not that th condition =C T is assumd hr to rprsnt a loss of load situation, which would b a consrvativ assumption. 2
22 On may think that, if d max >C T, thn LOLP=. Howvr, if F (d) is a tru probability distribution, thn it dscribs th vnt >C T with uncrtainty associatd with what th load is going to b, i.., only with a probability. On can tak an altrnativ viw, that th load duration curv is crtain, which would b th cas if w wr considring a prvious yar. In this cas, LOLP should b thought of not as a probability but rathr as th prcntag of tim during th intrval T for which th load quals or xcds capacity. It is of intrst to rconsidr (9), rpatd hr for convninc: t g( d) F ( d) P( d) T T (9) Substituting d=c T, w gt: F t g( CT ) ( CT ) P( CT ) T T (*) By (2), g(c T )=LOLE; by (3), P(>C T )=LOLP, and so (*) bcoms: LOLP LOLE T LOLE LOLPT which xprsss that LOLE is th xpctation of th numbr of tim units within T that dmand will xcd capacity. 22
23 Expctd dmand not srvd, ENS: If th avrag (or xpctd) dmand is givn by (), thn it follows that xpctd dmand not srvd is: ENS d max F C T ( ) d (4) which would b th sam ara as in Fig. 5 whn th ordinat is normalizd to provid F (d) instad of t. Rfrnc [3] provids a rigorous drivation for (4). Expctd nrgy not srvd, EENS: This is th total amount of tim multiplid by th xpctd dmand not srvd, i.., EENS T d max F which is th ara shown in Fig Effctiv load approach C T d max ( ) d g( ) d (5) Th notion of ffctiv load is usd to account for th unrliability of th gnration, and it is ssntial for undrstanding th viw takn in [3]. Th basic ida is that th total systm capacity is always C T, and th ffct of capacity outags ar accountd for by changing th load modl in an appropriat fashion, and thn th diffrnt indics ar computd as givn in (2), (3), (4), and (5). C T 23
24 A capacity outag of C i is thrfor modld as an incras in th dmand, not as a dcras in capacity! W hav alrady dfind as th random variabl charactrizing th dmand. Now w dfin two mor random variabls: is th random incras in load for outag of unit i. is th random load accounting for outag of all units and rprsnts th ffctiv load. Thus, th random variabls,, and ar rlatd: N (6) It is important to raliz that, whras C rprsnts th capacity of unit and is a dtrministic valu, rprsnts th incras in load corrsponding to outag of unit and is a random variabl. Th probability mass function (pmf) for is assumd to b as givn in Fig. 6 blow, i.., a twostat modl. W dnot th pmf for as f (d ). It xprsss th probability that th unit xprincs an outag of 0 MW as A, and th probability th unit xprincs an outag of C MW as U. 24
25 f (d ) A U 0 C Outag capacity, d Fig. 6: Two stat gnrator outag modl Rcall from probability thory that th pdf of th sum of two indpndnt random variabls is th convolution of thir individual pdfs, that is, for random variabls X and Y, with Z=X+Y, thn f Z ( z) f X ( z ) fy ( ) d (7) Similarly, w obtain th cdf of two random variabls by convolving th cdf of on of thm with th pdf (or pmf) of th othr, that is, for random variabls X and Y, with Z=X+Y, thn F Z ( z) FX ( z ) fy ( ) d (8) Lt s considr th cas for only unit, i.., from (6), Thn, by (8), w hav that: (9) 25
26 () (0) F ( d ) F ( d ) f ( ) d (20) ( ) whr th notation F ( ) indicats th cdf aftr th th unit is convolvd in. Undr this notation, thn, (9) bcoms ( ) and th gnral cas for (20) is: ( ) ( ) ( ) F ( d ) F ( d ) f ( ) d (2) (22) which xprsss th quivalnt load aftr th th unit is convolvd in. Sinc f (d ) is discrt (a pmf), w rwrit (22) as d ( ) F ( d ) F ( d d ) f ( d ) ( ) (23) From an intuitiv prspctiv, (23) is providing th ( ) convolution of th cdf F ( ) with th st of impuls functions comprising f (d ). Whn using a 2 stat modl for ach gnrator, f (d ) is comprisd of only 2 impuls functions, on at 0 and on at C. Rcalling that th convolution of a function with an impuls function simply shifts and scals that function, (23) can b xprssd for th 2stat gnrator modl as: F ( ) ( d ) A F ( ) ( d ) U F ( ) ( d C ) (24) 26
27 So th cdf for ffctiv load, following convolution with capacity outag pmf of th th unit, is th sum of th original cdf, scald by A and th original cdf, scald by U, rightshiftd by C. Exampl : Fig. 7 illustrats th convolution procss for a singl unit C =4 MW supplying a systm having pak dmand d max =4 MW, with dmand cdf givn as in plot (a) basd on a total tim intrval of T= yar. 0.8 * F r ( (0) d ) (a) f (d ) (b) C = d (c). (d) d d = F r ( () d ) () d Fig. 7: Convolving in th first unit 27
28 Plots (c) and (d) rprsnt th intrmdiat stps of (0) th convolution whr th original cdf F ( d ) was scald by A =0.8 and U =0.2, rspctivly, and rightshiftd by 0 and C =4, rspctivly. Not th ffct of convolution is to sprad th original cdf. Plot (d) may rais som qustion sinc it appars that th constant part of th original cdf has bn xtndd too far to th lft. Th rason for this apparnt discrpancy is that all of th original cdf, in plot (a), was not shown. Th complt cdf is illustratd in Fig. 8 blow, which shows clarly that (0) F ( d ) for d <0, rflcting th fact that P( >d )= for d < F r ( (0) d ) d Fig. 8: Complt cdf including valus for d <0 Lt s considr that th first unit w ust convolvd in is actually th only unit. If that unit wr prfctly rliabl, thn, bcaus C =4 and d max =4, our systm would nvr hav loss of load. This would b th 28
29 situation if w applid th idas of Fig. 5 to Fig. 7, plot (a). Howvr, Fig. 7, plot () tlls a diffrnt story. Fig. 9 applis th idas of Fig. 5 to Fig. 7, plot () to show how th cdf on th quivalnt load indicats that, for a total capacity of C T =4, w do in fact hav som chanc of losing load F r ( () d ) C T = d Fig. 9: Illustration of loss of load rgion Th dsird indics ar obtaind from (2),(3), (4): LOLE t g ( C ) T F ( C 4) yars CT T r T A LOLE of 0.2 yars is 73 days, a vry poor rliability lvl that rflcts th fact w hav only a singl unit with a high FOR=0.2. Th LOLP is givn by: LOLP P( C ) F ( C ) 0.2 T and th ENS is givn by: ENS d, max F ( ) d C T T 29
30 which is ust th shadd ara in Fig. 9, most asily computd using th basic gomtry of th figur, according to: 0.2() (3)(0.2) 0.5MW 2 Th EENS is givn by EENS T d,max F C T ( ) d d,max g C T ( ) d or TENS=(0.5)=0.5MWyars, or 8760(0.5)=4380MWhrs. Exampl 2: This xampl is from [4]. A st of gnration data is providd in Tabl 5. Tabl 5 Th 4 th column provids th forcd outag rat, which w hav dnotd by U. Th twostat 30
31 gnrator outag modl for ach unit is obtaind from this valu, togthr with th ratd capacity, as illustratd in Fig. 20, for unit. Notic that th units ar ordrd from last cost to highst cost. f (d ) A =0.8 U =0.2 0 C =200 Outag load, d Fig. 20: Twostat outag modl for Unit Load duration data is providd in Tabl 6 and plottd in Fig. 2. Tabl 6 3
32 Fig. 2 W now dploy (24), rpatd hr for convninc, F ( ) ( d ) A F ( ) ( d ) U F ( ) ( d C ) (24) to convolv in th unit outag modls with th load duration curv of Fig. 2. Th procdur is carrid out in an Excl sprad sht, and th rsult is providd in Fig. 22. In Fig. 22, w hav shown Original load duration curv, F0; Load duration curv with unit convolvd in, F. Load duration curv with all units convolvd in, F9 W could, of cours, show th load duration curvs for any numbr of units convolvd in, but this would b a cluttrd plot. 32
33 Fig. 22 W also show, in Tabl 7, th rsults of th calculations prformd to obtain th sris of load duration curvs (LC) F0F9. Notic th following: Each LC is a column FOF9 Th first column, in MW, is th load. o It bgins at 200 to facilitat th convolution for th largst unit, which is a 200 MW unit. o Although it xtnds to 2300 MW, th largst actual load is 000 MW; th xtnsion is to obtain th quivalnt load corrsponding to a 000 MW load with 300 MW of failabl gnration. Th ntris in th tabl show th % tim th load xcds th givn valu. LOLP is, for a particular column, th % tim load xcds th total capacity corrsponding to that column, and is undrlind. 33
34 For xampl, on obsrvs that LOLP= if w only hav units (F, C T =200) or only units and 2 (F2, C T =400). This is bcaus th capacity would nvr b nough to satisfy th load, at any tim. And LOLP= if w hav only units, 2, and 3 (F3, C T =600). This is bcaus w would b abl to supply th load for som of th tim with this capacity. And LOLP= if w hav all units (F9, C T =300), which is non0 (in spit of th fact that C T >000) bcaus units can fail. Tabl 7 34
35 5.0 Production cost modling using ffctiv load Th most basic production cost modl obtains production costs of thrmal units ovr a priod of tim, say yar, by building upon th quivalnt load duration curv dscribd in Sction 5. To prform this, w will assum that gnrator variabl cost, in $/MWhr, for unit oprating at P ovr a tim intrval t, is xprssd by C (E )=b E whr E =P t is th nrgy producd by th unit during th hour and b is th unit s avrag variabl costs of producing th nrgy (w omit fixd costs bcaus w ar only trying to quantify production costs hr). Th production cost modl bgins by assuming th xistnc of a loading (or mrit) ordr, which is how th units ar xpctd to b calld upon to mt th dmand facing th systm. W assum for simplicity that ach unit consists of a singl block of capacity qual to th maximum capacity. It is possibl, and mor accurat, to divid ach unit into multipl capacity blocks, but thr is no concptual diffrnc to th approach whn doing so. 35
36 Tabl 5, listd prviously in Exampl 2, provids th variabl cost for ach unit in th appropriat loading ordr. This tabl is rpatd hr for convninc. Tabl 5 Th critrion for dtrmining loading ordr is clarly conomic. Somtims it is ncssary to altr th conomic loading ordr to account for mustrun units or spinning rsrv rquirmnts. W will not considr ths issus in th discussion that follows. To motivat th approach, w introduc th concpt of a unit s obsrvd load as th load sn by a unit ust bfor it is committd in th loading ordr. Thus, it will b th cas that all highrpriority units will hav bn committd. If all highrpriority units would hav bn prfctly rliabl (A =), thn th obsrvd load sn by th 36
37 nxt unit would hav bn ust th total load lss th sum of th capacitis of th committd units. Howvr, all highrpriority units ar not prfctly rliabl, i.., thy may fail according to th forcd outag rat U. This mans w must account for thir stochastic bhavior ovr tim. This can b don in a straightforward fashion by using th quivalnt load duration curv dvlopd for th last unit committd. In th notation of (24) unit ss a load charactrizd ( by F ) ( d ). Thus, th nrgy providd by unit is ( proportional to th ara undr F ) ( d ) from x  to x, whr x  is th summd capacity ovr all prviously committd units and x is th summd capacity ovr all prviously committd units and unit. But unit is only going to b availabl A % of th ( tim. Also, sinc F ) ( d ) is a probability function, w must multiply it by T, rsulting in th following xprssion for nrgy providd by unit [5]: whr E TA x x F ( ) ( ) d (25) 37
38 x C i, x C i (26) i i Rfrring back to Exampl 2, w dscrib th computations for th first thr ntris. This dscription is adaptd from [4]. For unit, th original load duration curv F0 is usd, as forcd outags of any units in th systm do not affct unit l's obsrvd load. Th nrgy rqustd by th systm from unit, xcluding unit (0) l's forcd outag tim, is th ara undr F ( d ) ovr th rang of 0 to 200 MW (unit 's position in th loading ordr) tims th numbr of hours in th (0) priod (8760) tims A. Th ara undr F ( d ) from 0 to 200, illustratd in Fig. 23 blow, is 200. Fig
39 Thrfor, E ,40,600 MWhrs For unit 2, th load duration curv F is usd, as forcd outag of unit will affct unit 2's obsrvd load. Th nrgy rqustd by th systm from unit 2, xcluding unit 2's forcd outag tim, is th ara () undr F ( d ) ovr th rang of 200 to 400 MW (unit 2 's position in th loading ordr) tims th numbr of hours in th priod (8760) tims A 2. Th ara undr () F ( d ) from 200 to 400, illustratd in Fig. 24 blow, is 200. Thrfor, Fig. 24 E ,40,600 MWhrs 39
40 For unit 3, th load duration curv F2 is usd, as forcd outag of units and 2 will affct unit 3's obsrvd load. Th nrgy rqustd by th systm from unit 3, xcluding unit 3's forcd outag tim, is (2) th ara undr F ( d ) ovr th rang of 400 to 600 MW (unit 3 's position in th loading ordr) tims th numbr of hours in th priod (8760) tims A 3. Th (2) ara undr F ( d ) from 400 to 600, illustratd in Fig. 25, is calculatd blow Fig 25. Th coordinats on Fig. 25 ar obtaind from Tabl 7, rpatd on th nxt pag for convninc. (500,0.872) (600,0.66) Fig. 25 Th ara, indicatd in Fig. 25, is obtaind as two applications of a trapzoidal ara (/2)(h)(a+b), as 40
41 (00)(.872) (00)( ) 2 2 LftPortio n Thrfor, RightPortion E ,324,52 MWhrs Tabl 7 Continuing in this way, w obtain th nrgy producd by all units. This information, togthr with th avrag variabl costs from Tabl 5, and th rsulting nrgy cost, is providd in Tabl 8 blow. 4
42 Tabl 8 Unit MWhrs Avg. Variabl Enrgy Costs, $ Costs, $/MWhr,40, ,0,400 2,40, ,0,400 3,324, ,76, , ,823, ,00 58.,393,40 6 7, ,820, , ,724, , ,940, , ,856,480 Total E T = 5,289,300 99,54,280 It is intrsting to not that th total nrgy supplid, E T =5,289,300 MWhrs, is lss than what on obtaind whn th original load duration curv is intgratd. This intgration can b don by applying our trapzoidal approach to curv F0 in Tabl 7. oing so rsults in E 0 =5,299,800 MWhrs. Th diffrnc is E 0 E T =5,299,8005,289,300=0,500 MWhrs. What is this diffrnc of 0,500 MWhrs? To answr this qustion, considr: Th total ara undr th original curv F0, intgratd from 0 to 000 (th pak load), is 5,299,800 MWhrs, as shown in Fig. 26. This is th amount of nrgy providd to th actual load if it wr supplid by prfctly rliabl gnration having capacity of 000 MW. As indicatd abov, w will dnot this as E 0. 42
43 E 0 =8760*this ara =5,299,800 MWhrs Fig. 26 Th total ara undr th final curv, F9, intgratd from 0 to 300 MW (th gnration capacity) is E 300 =6,734,696 MWhrs, as shown in Fig. 27. This is th amount of nrgy providd to th ffctiv load if it wr supplid by prfctly rliabl gnration having capacity of 300 MW. E 300 = 8760*this ara =6,734,696 MWhrs Fig
44 Th nrgy rprsntd by th ara of Fig. 27, which is th nrgy providd to th ffctiv load if it wr supplid by prfctly rliabl gnration having capacity of 300 MW, is gratr than th nrgy providd by th actual 300 MW, that is E 300 >E T bcaus E 300 includs load rquird to b srvd whn th gnrators ar outagd, and this portion was xplicitly rmovd from th calculation of Tabl 8 (E T ). On can obsrv this radily by considring a systm with only a singl unit. Rcalling th gnral formula (25) for obtaining actual nrgy supplid by a unit pr th mthod of Tabl 8: E TA x x F ( ) ( ) d and applying this to th onunit systm, w gt: E T C (0) E TA F ( ) d 0 (25) (27) In contrast, th nrgy E obtaind whn w intgrat th ffctiv load duration curv (accounting for only th on unit) is 44
45 45 d F T E C 0 () ) ( (28) Rcalling th convolution formula (24), ) ( ) ( ) ( ) ( ) ( ) ( C d F U d A F d F (24) and for th onunit cas, w gt ) ( ) ( ) ( (0) (0) () C d F U d A F d F (29) Substituting (29) into (28) rsults in d C F U A F T E C 0 (0) (0) ) ( ) ( (30) Braking up th intgral givs 0 (0) 0 (0) 0 (0) 0 (0) ) ( ) ( ) ( ) ( C C C C d C F TU d F TA d C F U T d A F T E (3) Comparing (3) with (27), rpatd hr for convninc: d F TA E E C T 0 (0) ) ( (27) w obsrv th xprssions ar th sam xcpt for th prsnc of th scond intgration in (3). This provs that E >E T, i.., ffctiv nrgy dmandd > nrgy srvd by gnration
46 Now considr computing th nrgy consumd by th total ffctiv load as rprsntd by Fig. 28 (not that in this figur, th curv should go to zro at Load=2300 but dos not du to limitations of th drawing facility usd). E 2300 = 8760*this ara =6,745,200 MWhrs Fig. 28 Using th trapzoidal mthod to comput this ara rsults in E 2300 = MWhrs, which is th nrgy providd to th ffctiv load if it wr supplid by prfctly rliabl gnration having capacity of 2300 MW. This would lav zro nrgy unsrvd. Th diffrnc btwn o E 2300, th nrgy providd to th ffctiv load if it wr supplid by 2300 MW of prfctly rliabl gnration and o E 300, th nrgy providd to th ffctiv load if it wr supplid by 300 MW of prfctly rliabl gnration is givn by: E E 300 =6,745,2006,734,696=0,504 MWhrs 46
47 This is th xpctd nrgy not srvd (EENS), somtims calld th xpctd unsrvd nrgy (EUE). W obsrv, thn, that w can obtain EENS in two diffrnt approachs.. E 0 E T =5,299,8005,289,300=0,500 MWhrs 2. E E 300 =6,745,2006,734,696=0,504 MWhrs Approach may b computationally mor convnint for production costing bcaus E T is asily obtaind as th summation of all th nrgy valus. Approach 2 may b mor convnint concptually as it is simply th ara undr th ffctiv load curv from total capacity (I call it C T ) to infinity. 6.0 Commnts on W&W approach W&W, in sction 8.3.2, rfrs to th unsrvd load mthod. It is somwhat diffrnt from th ffctiv load mthod dscribd abov. Th main diffrnc can b obsrvd by comparing quation (8.2) in your txt with quation (24) usd abov. 47
48 F ( ) ( d ) A F ( ) ( d ) U F ( ) ( d C ) (24) Pn ( x) qpn ( x) ppn ( x C) (8.2) Both lfthand xprssions ar th nw cdf aftr convolving in a unit. Spcifically, th nomnclatur rlats as follows: d =x (valu of quivalnt load) C =C (capacity of unit ) A =p (availability of unit ) U =q (unavailability of unit ) On obsrvs that th two quations ar almost th sam, with two xcptions:. Shift: Whras th shift on th scond trm of (24) is a rightshift by an amount C, th shift on th scond trm of (8.2) is a lftshift by an amount C. 2. A and U : Whras th unshiftd (first) trm of (24) is multiplid by A =p, th unshiftd (first) trm of (8.2) is multiplid by U =q. Th diffrnc should b undrstood. 48
49 Whras th ffctiv load mthod o xtnds or incrass th load to probabilistically account for gnrator unavailability, o and uss total capacity undr assumption of prfct rliability to assss mtrics th unsrvd load mthod o rducs or dcrass th load to probabilistically account for gnrator availability, o and uss zro capacity to assss mtrics. 6.0 W&W (unsrvd load) mthod W will maintain th notation usd in dscribing th ffctiv load mthod. Th diffrncs in notation rlativ to W&W ar dscribd in th prvious sction. fin as th random variabl charactrizing th dmand. Now w dfin two mor random variabls: is th random dcras in load for (probabilistic) availability of unit. is th random load accounting for th (probabilistic) availability of all units and rprsnts th unsrvd load. 49
50 Thus, th random variabls,, and ar rlatd: N (25) Whras C rprsnts th capacity of unit and is a dtrministic valu, rprsnts an ffctiv dcras in load corrsponding to (probabilistic) availability of unit and is a random variabl. Th probability mass function (pmf) for is assumd to b as givn in Fig. 29, i.., a twostat modl. W dnot th pmf for as f (d ). It xprsss th probability that th unit xprincs an outag of 0 MW as A, and th probability th unit xprincs an outag of C MW as U. f (d ) A U 0 C Availabl capacity, d Fig. 29: Two stat gnrator availability modl W saw in th abov nots that th pdf of th sum of 2 indpndnt random variabls is th convolution of thir individual pdfs, that is, for random variabls X and Y, with Z=X+Y, thn 50
51 f Z ( z) f X ( z ) fy ( ) which can also b writtn as: f d Z ( z) f X ( ) fy ( z ) d (7) Likwis, th pdf of th diffrnc of 2 indpndnt random variabls is also a convolution, that is, for random variabls X and Y, with Z=XY, thn f Z ( z) f X ( z) fy ( z ) d (26) In addition, it is tru that th cdf of th diffrnc btwn 2 random variabls can b found by convolving th cdf of on of thm with th pdf (or pmf) of th othr, that is, for random variabls X and Y, with Z=XY, thn F Z ( z) FX ( z) fy ( z ) d (27) Lt s considr th cas for only unit, i.., from (25), Thn, by (27), w hav that: (28) 5
52 F (0) ( d) F ( ) f ( d ) d () (20) ( ) whr th notation F ( ) indicats th cdf aftr th th unit is convolvd in. With this notation, (28) is ( ) ( ) and th gnral cas for (29) is: F ( ) ( ) ( d) F ( ) f ( d ) d (29) (30) which xprsss th quivalnt load aftr th th unit is convolvd in, considring th (probabilistic) availability of unit and all lowr numbrd units. Sinc f (d ) is discrt (a pmf), w rwrit (30) as d ( ) F ( d ) F ( d ) f ( d d ) ( ) (3) From an intuitiv prspctiv, (3) is providing th ( ) convolution of th cdf F ( ) with th st of impuls functions comprising f (d ). Whn using a 2 stat availability modl for ach gnrator, f (d ) is comprisd of only 2 impuls functions, on at 0 and on at C. Rcalling that th convolution of a function with an impuls function simply shifts and scals that function, (3) can b xprssd for th 2stat gnrator modl shown in Fig. 23 as: 52
53 F ( ) ( d ) U F ( ) ( d ) A F ( ) ( d C ) (32) So th cdf for th ffctiv load, following convolution with capacity outag pmf of th th unit, is th sum of th original cdf, scald by U and th original cdf, scald by A, lftshiftd by C. W&W say this (p. 287): Th first trm is th probability that nw capacity C is unavailabl tims th probability of nding an amount of powr d or mor; Th scond trm is th probability C is availabl tims th probability d +C or mor is ndd. Exampl 3: Fig. 30 illustrats th convolution procss for a singl unit C =4 MW supplying a systm having pak dmand d max =4 MW, with dmand cdf givn as in plot (a) basd on a total tim intrval of T= yar. 53
54 0.8 * F r ( (0) d ) (a) f (d ) A =0.8 (b) U = d (c) (d) = Srvd load d d F r ( () d ) Unsrvd load () d Fig. 30: Convolving in th first unit (not prfctly rliabl) Plots (d) and (c) rprsnt th intrmdiat stps of (0) th convolution whr th original cdf F ( d ) was scald by U =0.2 and A =0.8, rspctivly, and lftshiftd by 0 and C =4, rspctivly. Not th ffct of convolution is to shift th original cdf to th lft. 54
55 Plot (c) may rais som qustion sinc it appars that th constant part of th original cdf has bn xtndd too far to th lft. Th rason for this is that all of th original cdf, in plot (a), was not shown. Th complt cdf is illustratd in Fig. 3 blow, which (0) shows clarly that F ( d ) for d <0, rflcting th fact that P( >d )= for d < F r ( (0) d ) d Fig. 3: Complt cdf including valus for d <0 Lt s considr that th first unit w ust convolvd in is actually th only unit. If that unit wr prfctly rliabl, thn, bcaus C =4 and d max =4, our systm would nvr hav loss of load. In this cas, with A = and U =0, th convolution procss abov would hav rsultd in Fig. 32 blow. Th fact that th final load duration curv F () (d ) shows Pr(d >0)=0 mans that thr is no chanc w will ncountr a load intrruption for this systm! 55
56 0.8 * F r ( (0) d ) (a) f (d ) A =.0 (b) U = d (c) (d) = Srvd load d d F r ( () d ) Unsrvd load () d Fig. 32: Convolving in th first unit (prfctly rliabl) Howvr, Fig. 30, plot () tlls a diffrnt story. Th fact that thr is som part of th load duration curv to th right of d =0 is an indication that thr is a possibility of load intrruption. 56
57 Obsrv that positiv d may b thought of as unsrvd load; ngativ d may b thought of as srvd load. In othr words, Fig. 32 tlls us Pr(unsrvd load > 0 MW) = 0 Pr(unsrvd load > 4 MW)=.0 Pr(srvd load < 4 MW)=.0 Fig. 30 applis th idas of Fig. 5 to Fig. 30, plot () to show how th cdf on th quivalnt load indicats that, for a total capacity of C T =0, w do in fact hav som chanc of losing load F r ( () d ) d Fig. 33: Illustration of loss of load rgion Th dsird indics ar obtaind from (2),(3), (4): LOLE t C g( CT ) T F ( CT 0) yars T r A LOLE of 0.2 yars is 73 days, a vry poor rliability lvl that rflcts th fact w hav only a singl unit with a high FOR=0.2. Th LOLP is givn by: 57
58 LOLP P( C ) F ( C ) 0.2 and th ENS is givn by: ENS d T, max F ( ) d C T which is ust th shadd ara in Fig. 33, most asily computd using th basic gomtry of th figur, according to: 0.2() (3)(0.2) 0.5MW 2 Th EENS is givn by EENS T d,max F C T ( ) d d,max g C T T ( ) d or TENS=(0.5)=0.5MWyars, or 8760(0.5)=4380MWhrs. Exampl 4: This xampl is from [6]. A st of gnration data is providd in Tabl 8. Tabl 8 58
59 Obsrv th units ar ordrd from last to highst cost. Th 4 th column provids th forcd outag rat (FOR), which w hav dnotd by U. Th twostat gnrator outag modl for ach unit (obtaind from th FOR), togthr with th ratd capacity, is illustratd in Fig. 34, for unit. f (d ) A =0.8 U =0.2 0 C Availabl capacity, d Fig. 34: Twostat outag modl for Unit Load duration data is providd in Tabl 9 and plottd in Fig. 35. Tabl 9 59
60 Fig. 35 W now dploy (32), rpatd hr for convninc, F ( ) ( d ) U F ( ) ( d ) A F ( ) ( d C ) (32) to convolv in th unit outag modls with th load duration curv of Fig. 35. Th procdur is carrid out in an Excl sprad sht, and th rsult is providd in Fig. 36. In Fig. 36, w hav shown Original load duration curv, F0; Load duration curvs with unit convolvd in, F, =,,9. 60
61 Fig. 36 W also show, in Tabl 0, th rsults of th calculations prformd to obtain th sris of load duration curvs (LC) F0F9. Notic th following: Each LC is a column FOF9 Th first column, in MW, is th load. o It bgins at 400, an arbitrarily chosn larg ngativ numbr to nsur ach LC bgins from th lft with an ordinat of.0 (w rally only nd to xtnd to 900). o Th largst actual load is 000 MW; th xtnsion is to obtain th quivalnt load corrsponding to a 000 MW load with 300 MW of failabl gnration. Th ntris in th tabl show th % tim th unsrvd load xcds th givn valu. LOLP is, for a particular column, th % tim load xcds th total capacity corrsponding to that column, and is undrlind. 6
62 For xampl, on obsrvs that LOLP= if w only hav units (F, C T =200) or only units and 2 (F2, C T =400). This is bcaus th capacity would nvr b nough to satisfy th load, at any tim. And LOLP= if w hav only units, 2, and 3 (F3, C T =600). This is bcaus w would b abl to supply th load for som of th tim with this capacity. And LOLP= if w hav all units (F9, C T =300), which is non0 (in spit of th fact that C T >000) bcaus units can fail. Tabl 0 F0 F F2 F3 F4 F5 F6 F7 F8 F Load (MW) Fraction of tim load xcds givn load E E05.25E E05.0E05.62E E E06.8E06.3E E E E E07 2E08 2E09 E
63 7.0 Production cost modling using unsrvd load Th most basic production cost modl obtains production costs of thrmal units ovr a priod of tim, say yar, by building upon th procdurs dscribd in Sction 7. Th production cost modl bgins by assuming th xistnc of a loading (or mrit) ordr, which is how th units ar xpctd to b calld upon to mt th dmand facing th systm. W assum for simplicity that ach unit consists of a singl block of capacity qual to th maximum capacity. It is possibl, and mor accurat, to divid ach unit into multipl capacity blocks, but thr is no concptual diffrnc to th approach whn doing so. Tabl 5, listd prviously in Exampls 2 and 4, provids th variabl cost for ach unit in th appropriat loading ordr. This tabl is rpatd hr for convninc. 63
64 Tabl 5 Th critrion for dtrmining loading ordr is clarly conomic. Somtims it is ncssary to altr th conomic loading ordr to account for mustrun units or spinning rsrv rquirmnts. W will not considr ths issus in th discussion that follows. To motivat th approach, w introduc th concpt of a unit s obsrvd load as th load sn by a unit ust bfor it is committd in th loading ordr. Thus, it will b th cas that all highrpriority units will hav bn committd. If all highrpriority units would hav bn prfctly rliabl (A =), thn th obsrvd load sn by th nxt unit would hav bn ust th total load lss th sum of th capacitis of th committd units. 64
65 Howvr, all highrpriority units ar not prfctly rliabl, i.., thy may fail according to th forcd outag rat U. This mans w must account for thir stochastic bhavior ovr tim. This can b don in a straightforward fashion by using th quivalnt load duration curv dvlopd for th last unit committd. In th notation of (32) unit ss th unsrvd load ( charactrizd by F ) ( d ). Thus, th nrgy providd ( by unit is proportional to th ara undr F ) ( d ) from 0 to C, whr C is th capacity of unit. In our xampl 4 abov, Unit ss th ntir load, (0) charactrizd by F ( d ), illustratd as th whit ara in Fig. 37. Fig
66 Unit 2, howvr, will s th load aftr unit has bn convolvd in (rsulting in F), which will hav th ffct of rducing th unsrvd load, illustratd in th whit ara in Fig. 38 (which is lss ara than th whit ara in Fig. 37). Fig. 38 W want to comput th cost of running ach of th various units. W assum that gnrator cost rat, in $/hr, for unit oprating at P, is linarizd, xprssd by J =J 0 +J P (33) whr J 0 is th unit s noload cost rat, $/hr and J is th unit s cost of nrgy production, $/MWhr. W also assum that unit has capacity C. 66
67 W obtain th cost associatd with schduling unit according to (33). Considration of th noload costs is asy, bcaus w will incur thm for vry hour th unit is schduld. Lt s assum that th unit will b schduld for th ntir tim priod, T hours, whr T is th numbr of hours charactrizd by th CF ( ) F ( d ). Thrfor th noload costs is (in $): NoloadCosts=J 0 T (34) Th variabl (ful) costs could b computd (in $) as: VariablCosts=J P T Howvr, this would rquir that th unit runs at P for all T hours. That may not happn. A bttr way to comput variabl costs rsults from rcognizing that J, with units of $/MWhr, is th cost pr unit of nrgy. Thrfor, if w can gt th nrgy supplid by unit, E, thn this will allow us to comput th cost of supplying it from VariablCosts=J E (35) Th nrgy supplid by unit can b computd from ( th CF F ) ( d ) according to th following: E T C 0 F ( ) ( ) d (36) In our xampl, for unit (a 200 MW unit), this would corrspond to th ara dnotd by th hatchd rgion in Fig
68 Fig. 39 In this particular cas, th intgration of (36) provids th sam answr as P T, but this is bcaus this unit is basloadd and dos in fact run all tim at capacity. And so th total cost of schduling unit can b valuatd as th sum of th noload and variabl costs, which is: TotalCosts=NoloadCosts+VariablCosts=J 0 T+ J E (37) whr E is givn by (36). Thr is ust on problm with (37) 68
69 Onc w commit a unit, w do intnd that it will b schduld for all T hours, Howvr, bcaus th unit has an availability of A, w can only xpct that th unit will b availabl for a numbr of hours qual to A T, i.., unit is only going to b availabl A % of th tim. Thrfor, w nd to modify (37) to b TotalCosts=A J 0 T+ A J E (38) whr, as bfor, E is givn by (36). Rfrring back to Exampl 4, w dscrib th computations for th first thr ntris. This dscription is adaptd from [4]. For unit, th original load duration curv F0 is usd, as forcd outags of any units in th systm do not affct unit l's obsrvd load. Th nrgy rqustd by th systm from unit is th ara undr (0) F ( d ) ovr th rang of 0 to 200 MW (unit s capacity) tims th numbr of hours in th priod (0) (8760) tims A =0.8. Th ara undr F ( d ) from 0 to 200, has alrady bn illustratd in Fig. 37 abov, and is 200. Thrfor, E ,40,600 MWhrs and th total cost of unit is 69
70 For unit 2, th load duration curv F is usd, as forcd outag of unit will affct unit 2's obsrvd load. Th nrgy rqustd by th systm from unit 2 () is th ara undr F ( d ) ovr th rang of 0 to 200 MW (unit 2 s capacity) tims th numbr of hours in th priod (8760) tims A 2 =0.8. Th ara undr () F ( d ) from 0 to 200, illustratd in Fig. 40 blow, is 200. Thrfor, Fig. 40 E ,40,600 MWhrs 70
71 For unit 3, th load duration curv F2 is usd, as forcd outag of units and 2 will affct unit 3's obsrvd load. Th nrgy rqustd by th systm (2) from unit 3 is th ara undr F ( d ) ovr th rang of 0 to 200 MW (unit 3 s capacity) tims th numbr of hours in th priod (8760) tims A 3 =0.9. Th ara (2) undr F ( d ) from 0 to 200, illustratd in Fig. 4, is calculatd blow. Th coordinats on Fig. 4 ar obtaind from Tabl 0, rpatd on th nxt pag for convninc. (00,0.872) (20,0.66) Fig. 4 Th ara, indicatd in Fig. 4, is obtaind as two applications of a trapzoidal ara (/2)(h)(a+b), as (00)(.872) (00)( ) 2 2 LftPortio n RightPortion 7
72 Thrfor, E ,324,52 MWhrs Tabl 0 F0 F F2 F3 F4 F5 F6 F7 F8 F Load (MW) Fraction of tim load xcds givn load E E05.25E E05.0E05.62E E E06.8E06.3E E E E E07 2E08 2E09 E Continuing in this way, w obtain th nrgy producd by all units. This information, togthr with th avrag variabl cost for ach unit from Tabl 5, and th rsulting variabl cost for ach unit, is providd in Tabl blow. Obsrv that in Tabl, th noload costs ar all zro, and so th total costs ar th sam as th variabl costs. 72
73 Unit i Noload cost cofficint J 0i ($/hr) Noload costs Tabl Enrgy E i Variabl cost cofficint J i ($/MWhr) Variabl Cost Total costs, J 0i *T+ J i E i ($) J 0i *T ($) (MWhr) J i E i ($) 0 0,40, ,0,400 9,0, ,40, ,0,400 9,0, ,324, ,76,500 35,76, , ,823,400 9,823, ,00 58.,393,40,393, , ,820,940 6,820, , ,724,20 3,724, , ,940,540,940, , ,856,480,856,480 Total 0 0 E T = 5,289,300 99,54,280 99,54,280 It is intrsting to not that th total nrgy supplid, E T =5,289,300 MWhrs, is lss than what on obtains whn th original load duration curv is intgratd. This intgration can b don by applying our trapzoidal approach to curv F0 in Fig. 37, rpatd hr for convninc, to obtain th whit ara shown in th figur. 73
74 E 0 =8760*this ara =5,299,800 MWhrs Fig. 37 oing so rsults in E 0 =5,299,800 MWhrs. Th diffrnc is E 0 E T =5,299,8005,289,300=0,500 MWhrs. What is this diffrnc of 0,500 MWhrs? To answr this qustion, considr th load duration curv aftr th last unit has bn convolvd in, curv F9, as shown in Fig. 42. Th total ara undr th original curv F0, intgratd from 0 to 000 (th pak load), is 5,299,800 MWhrs, as shown in Fig. 37. This is th amount of nrgy providd to th actual load if it wr supplid by prfctly rliabl gnration having capacity of 000 MW. As indicatd abov, w will dnot this as E 0. 74
75 Th total ara undr th final curv, F9, intgratd from 300 (th total srvd load) to 0 (th gnration capacity) is E s =6,734,696 MWhrs, as shown in Fig. 42. This is th amount of nrgy providd to th ffctiv load if it wr supplid by prfctly rliabl gnration having capacity of 300 MW. It is th srvd load. E s = 8760*this ara =6,734,696 MWhrs Fig. 42 Th nrgy rprsntd by th ara of Fig. 42, which is th nrgy providd to th ffctiv load if it wr supplid by prfctly rliabl gnration having capacity of 300 MW, is gratr than th nrgy providd by th actual 300 MW, that is E s > E T bcaus E s includs load rquird to b srvd whn th gnrators ar outagd, and this portion was xplicitly rmovd from th calculation of Tabl (E T ). On can obsrv this radily by 75
76 F considring a systm with only a singl unit. Combining th rlations (36) and (38), w can obtain th actual nrgy supplid by a unit (sam as mthod of Tabl ): E TA C 0 F ( ) ( ) d and applying this to th onunit systm, w gt: E T C (0) E TA F ( ) d 0 (39) (40) In contrast, th nrgy srvd E s obtaind whn w intgrat th ffctiv load duration curv (accounting for th on unit) is E s T 0 C F () ( ) d Rcalling th convolution formula (32), ( ) ( d ) U F ( ) ( d ) A F ( ) ( d C ) (4) (42) and for th onunit cas, w gt () (0) (0) F ( d) UF ( d) A F ( d C) (43) Substituting (43) into (4) rsults in E s T 0 C U F (0) ( ) A F (0) Braking up th intgral givs ( C ) d (44) 76
77 E s T 0 C U TU F 0 C (0) F ( ) d T (0) 0 C ( ) d TA A F 0 C (0) F (0) ( C ( C ) d ) d (45) Rvrsing th ordr of intgration and multiplying by  provids: C C (0) (0) E s TU F ( ) d TA F ( C) d (46) 0 0 Comparing (46) with (40), rpatd hr for convninc: E T C (0) E TA F ( ) d 0 (40) w obsrv th xprssions ar th sam xcpt for th prsnc of th scond intgration in (45). This provs that E s > E T Now considr computing th nrgy consumd by th total ffctiv load, which includs th unsrvd load, as rprsntd by Fig
78 E T = 8760*this ara =6,745,200 MWhrs Fig. 43 Using th trapzoidal mthod to comput this ara rsults in E T = MWhrs, which is th nrgy providd to th ffctiv load if it wr supplid by prfctly rliabl gnration having capacity of 2300 MW. This would lav zro nrgy unsrvd. Th diffrnc btwn o Total ffctiv load, E T : th nrgy providd to th ffctiv load if it wr supplid by 2300 MW of prfctly rliabl gnration and o Effctiv load srvd, E s : th nrgy providd to th ffctiv load if it wr supplid by 300 MW of prfctly rliabl gnration is givn by: E T E s =6,745,2006,734,696=0,504 MWhrs This is th xpctd nrgy not srvd (EENS), somtims calld th xpctd unsrvd nrgy (EUE). 78
79 W obsrv, thn, that w can obtain EENS in two diffrnt approachs.. E 0 E T =5,299,8005,289,300=0,500 MWhrs whr E 0 is th total nrgy dmandd by th actual load as computd from th original load duration curv; E T is th nrgy srvd to th actual load by th 300 MW of gnration accounting for ach unit s potntial to fail. 2. E T E s =6,745,2006,734,696=0,504 MWhrs whr E T is th total nrgy dmandd by th ffctiv load as computd from th complt ffctiv load duration curv; E s is th nrgy srvd to th ffctiv load by th 300 MW of gnration, assuming th 300 MW is prfctly rliabl. Approach may b computationally mor convnint for production costing bcaus E T is asily obtaind as th summation of all th nrgy valus. Approach 2 may b mor convnint concptually as it is simply th ara undr th ffctiv load curv from 0 to total capacity (w can call it C T ). 79
80 8.0 Additional W&W commnts of intrst A fw othr commnts about th W&W txt: Pg. 283: In rality, EENS is nrgy that would not b not srvd but rathr providd via xpnsiv intrconnction or mrgncy backup (providing nrgy via intrconnction was th original motivation bhind intrconncting control aras). Pg. 286: An altrnativ mthod of handling EENS is to plac mrgncy sourcs of vry larg capacity and high cost at th nd of th priority list, so that thy only gt usd if no othr capacity is availabl. Pg. 284: Mntions NERC s databas. It is calld GAS (Gnrating Availability ata Systm). Thr is also a TAS (Transmission Availability ata Systm) and a AS (mand Rspons Availability ata Systm). Pg. 284: For vry larg systms, th convolution mthod dscribd abov can b computationally intnsiv. An altrnativ mthod is calld th mthod of cumulants. Pg. 38: All of what w hav dscribd also applis whn gnrators ar modld as multistat dvics. This can account for th possibility of drating a unit which somtims occurs whn th unit rquirs a forcd rduction in output du to som particular part of th plant bcoming dysfunctional (.g., on out of 6 boilr fdpumps gos down). 80
81 9.0 Industrygrad commrcial production cost modls In th prvious nots, w rviwd a rlativly simpl production cost modl (PCM). This PCM rquird two basic kinds of input data: Annual load duration curv Unit data: o Capacity o Forcd outag rat o Variabl costs It thn computs load duration curvs for ffctiv load (which accounts for th unrliability of th gnrators supplying that load) through a convolution procss and provids th following information: Rliability indics: LOLP, LOLE, ENS, EENS (EUE) Annual nrgy producd by ach unit Annual production costs for ach unit Total systm production costs Anothr approach to PCMs is to simulat ach hour of th yar. This allows much mor rigorous modls and mor rfind rsults, which coms with a significant computational cost. Promod is on such modl which you will har about. I will dscrib th concptual approach to such PCMs. 8
82 9. A rfind production cost modl This PCM consists of th following loops:. Annual loop: Most PCMs hav only on annual loop, i.., th annual simulation is dtrministic. But it is concivabl to mak multipl runs through a particular yar, ach tim slcting various variabls basd on probability distributions for thos variabls. Such an approach is rfrrd to as a Mont Carlo approach, and it rquirs many loops in ordr to convrg with rspct to th avrag annual production costs. 2. UC loop: Th program must hav a way for dciding, in ach hour, which units ar committd. A UC program could b implmntd within th PCM on a wkly basis, a 48 hour basis, or a dayahad basis. Th lattr sms to b th prfrrd approach today bcaus it is consistnt with th fact that most lctricity markt structurs today dpnd on th dayahad using th scurityconstraind unit commitmnt. 3. Hourly loop: A scurity constraind optimal powr flow (SCOPF) is implmntd to dispatch availabl units. In addition, it is within th hourly loop that rliability indics ar computd. Thr ar two ways of doing this. Both ways dpnd on th fact that th load is dtrministic during th hour and so is rprsntd by a singl numbr. Th only randomnss is in rgards to th status of 82
83 committd gnrators and whthr thy ar in srvic or out of srvic du to a forcd outag. Mont Carlo: Status of ach committd gnrator is idntifid via random draw of a numbr btwn 0. If a numbr btwn 0 and th probability of th unit bing down (.g., 00.03) is chosn, th unit is outagd. If a numbr btwn probability of unit bing down & is chosn (.g., 0.03), th unit is in up. Analytic: A convolution mthod similar to our ffctiv load duration approach is mployd to comput rliability indics for th hour. Th mthod is simplr bcaus th load is dtrministic. Th mthod is rfrrd to as a capacity outag tabl approach; I can provid you with nots on this mthod if you want thm. Ntwork flows: This approach can also handl probabilistic tratmnt of transmission. Commnt: It is important to us outag rplacmnt rat (ORR) as th probability of th unit bing down, rathr than th forcd outag rat (FOR). Th ORR is th probability that th unit will go down in th nxt hour givn it is up at th bginning of th hour. 9.2 A rportd modl A modl is rportd in [7] which capturs som of th abov attributs. I hav liftd out two of th flow charts from this rfrnc to illustrat. 83
84 84
85 MTH 40,000 20,000 00,000 80,000 60,000 40,000 20,000 0 Fb03 Mar03 Wintr Apr03 May03 COH  Firm Rquirmnts and Supply Jul03 Aug03 Sp03 Oct03 Summr 2003 Annual Nov03 Endusr Balancing Exchang Payback Inctions Firm mand Endusr Supply/Balancing Exchang Supply Withdrawals Purchass 0.0 MISO s us of Production Costing Blow ar a fw mor slids that charactriz how MISO utilizs production costing. Background PROMO is a Production Cost Modl dvlopd by Vntyx (Formrly known as NwEnrgy Associats, A Simns Company). taild gnrator portfolio modling, with both rgion zonal pric and nodal LMP forcasting and transmission analysis including marginal losss 30 How PROMO Works  PROMO Structur PSS/E Rport Agnt Visualization & Rporting Common API Common ata Sourc  taild unit commitmnt and dispatch  taild transmission simulation  Asst Valuation with MarktWis  FTR Valuation with TAM  Easytous intrfac  Powrful scnario managmnt  Complt NERC data with solvd powrflow cass Accss, Excl, Pivot Cub 3 85
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